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week12tutsols

# week12tutsols - THE UNIVERSITY OF SYDNEY Math2968...

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THE UNIVERSITY OF SYDNEY Math2968 Algebra (Advanced) Semester 2 Tutorial Solutions Week 12 2008 1. (for general discussion) Recall that a poset L is a lattice if x y = inf { x,y } and x y = sup { x,y } exist for all x,y L . Discuss the existence or otherwise of each of the following: inf { x } , sup { x } , inf , sup , inf L, sup L Solution Both the following always exist for x L : inf { x } = inf { x,x } = x x = x and sup { x } = sup { x,x } = x x = x The largest and smallest elements of L may or may not exist and, when they do, equal sup L = inf and inf L = sup respectively. 2. Let L be a lattice and choose x,y L such that x y . Verify that the interval [ x,y ] = { z L | x z y } is a sublattice of L . Solution By reFexivity, x x y , so x [ x,y ] n = . Let a,b [ x,y ], so x a y and x b y . In particular, y is an upper bound for a and b . But a b is the least upper bound, so a b y . In particular also, a b is an upper bound for a , so x a a b y . By transitivity, x a b y , so that a b [ x,y ]. Dually (interchanging with ) we get that a b [ x,y ]. Hence [ x,y ] is closed under join and meet so is a sublattice. 3. Verify the following associativity law for joins in a lattice: x ( y z ) = ( x y ) z Write down the corresponding associativity law for meets and explain how it follows by duality. Solution Put α = x ( y z ) and β = ( x y ) z . Then α is an upper bound for x and y z , which in turn is an upper bound for y and z . By transitivity, α is an upper bound for x , y and z . In particular, x y α , since x y is the least upper bound for

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week12tutsols - THE UNIVERSITY OF SYDNEY Math2968...

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